What is Happening with CAS in Classrooms Example Austria by nrk14057

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									                      What is Happening with CAS in Classrooms?
                                   Example Austria
                                           Josef Böhm, ACDCA, Austria
            Abstract: In this paper a short overview is given how CAS is used in Austrian classrooms.
            The main part deals – according to the wishes of the organizers of this CAME strand – with
            real classroom examples and experiences. They are embedded in a kind of framework to
            explain the status of CAS-supported learning, teaching and assessing in Austria. I left most
            of the examples in their original form (in German) to stay as authentic as possible. This
            overview does not pretend to be complete – it is my personal sight of the matter.


1        CAS in Austria
Introduction and propagating the use of CAS in math education is in a very tight connection with the
Austrian Centre for Didactics in Austria which was founded by school authorities and enthusiastic teachers
in the early nineties. The ACDCA is hosted by the Pedagogical Institute of Lower Austria in Hollabrunn.
1991 the Austrian Government purchased a general licence of DERIVE for all general secondary school
(grammar schools). The ACDCA was founded with the intention to attend and support teachers in using this
revolutionary tool in a meaningful and responsible way unlike to the introduction of the pocket calculator
several years before.
There was a fruitful cooperation with the T3-organization for several years. Useful synergy-effects made pre-
and in-service trainings very successful. Unfortunately this cooperation was ended by Texas Instruments in
2005. It is a fact that Austria – seen as a whole - was never a graphing calculator country, which was caused
by the early use of DERIVE.

The ACDCA initiated and supervised a couple of Austrian wide projects:

                                         Austrian ACDCA Projects
               Austrian Centre for the Didactics of Computer Algebra, PI Lower Austria, Hollabrunn

CAS I       1993 – 1994           DERIVE Project
CAS II      1997 – 1998          TI-92 Project
            Creating teaching materials – Influence of CAS on teaching
            44 schools, 70 classes (65 teachers & 680 female, 1570 male students) involved
CAS III     1999 – 2000         2nd TI-92 Project
            Electronic Learning Media in Maths Education
            Influence on Teaching, Learning, Curriculum and Assessment
            94 classes with more than 2000 students involved
CAS IV      2001 – 2002         CAS Project
            New Media and Methods (New culture of problems, Supervision,
            Bilingual Teaching, Establishing a Service Centre for Teachers, …
            140 classes with more than 2200 students involved
CAS V       2003 – 2005        Variety of Media Project
            together with GeoGebra and Mathe-Online
            e- & Online-Learning, Self responsible Learning,
            Standards for Mathematics, Teaching in “Laptop classes”, …
CAS VI      in preparation

The most used CAS in Austria is DERIVE (General Secondary schools and Secondary Colleges for Business
Administration, followed by the handheld CAS (TI92/TI89/V200). The Secondary Colleges for Engineering
use MathCAD (general licence) and the Voyage 200. There is a minority using MATHEMATICA and
MuPAD. Other Secondary Vocational Schools use – if they use CAS – handheld CAS or DERIVE.


Josef Böhm: CAS in Austrian Classrooms                                                                     1
2       CAS and Curriculum
Curricula demand the use of technology for all secondary schools. The respective part in the curriculum for
general secondary schools reads:

Technologies like computer algebra systems, spreadsheet programs, dynamic geometry software shall play
an important role in mathematics education. Students shall be able to use these programs up to an amount
which is relevant for the mathematical content. Minimum realisation is introducing these technologies not
only for some examples but using them for acquiring some fundamental concepts. Maximum realisation is
the meaningful use of them as an integrative and consequent part of the teaching and learning process.

This is the respective paragraph of the curriculum of Secondary Colleges of Business Administration:

Die Schülerinnen und Schüler sollen
    …..
    - Computer Algebra Systeme und/oder Tabellenkalkulation bzw. grafikfähige Taschenrechner in allen
        Jahrgängen einsetzen und mathematische Problemstellungen damit lösen können.
The students shall be able to
   …..
   - use Computer Algebra Systems and/or Spreadsheet programs or graphing calculators respectively in all
       forms and to solve mathematical problems by using these tools.

The Secondary Colleges for Engineering have been using CAS for a long time (MathCAD and TI-CAS).

The Austrian school system provides freedom in teaching methods. As there are no central final exams it is
in the responsibility of the teachers if and in which amount CAS is permitted or even demanded – or if it is
completely banned.

3       CAS and Assessment
If CAS is used in learning and teaching then it is also used for assessment. I show some examples from my
teacher career. Problems and questions changed compared with traditional assessment problems. All the
following problems had not been trained. The students saw problems like these during the test the first time.
(Form 10 = age 16, form 11 = age 17)

Example 1 (form 10)
Solve the simultaneous equations for x and y and find at least two pairs (a,b), which make the system non
soluble!
                                       a x + b y = a + 2 y − 3x
                                         2a x − 3b y = b − 3 y + 2 x
This problem was given in times when the TI-92 was not able so solve systems of linear equations with one
solve-command. I expected the students to bring the unknown and the parameters in a right order and apply
any solution algorithm (Cramer´s Rule, Row Echelon Form, …) and then find out two pairs (a,b) which
equate the denominator (see the TI-92 screen shot):

(a + 3) x + (b – 2) y = a
(2a -–2) x + (3 – 3b) y = b

For example a = 5 and b = 5/4 form one such pair.



I was very much surprised that one girl was ready within a minute – without using the calculator. I thought
that she was cheating and asked her what she did: ‘Oh’, she said, ‘that´s easy. Take the first line: a = -3 and
b = 2 make a contradiction 0 = -3. This is the first pair. The second line gives the other one.’ This was one of
the – many moments – when I was happy to be a math teacher.

Josef Böhm: CAS in Austrian Classrooms                                                                         2
Example 2 (form 10)
a) Enter the equation and explain the TI´s "simplification".
b) What is the equation´s general solution and which conditions are necessary for its validity?
c) Find two special solutions!
d) What is the result of the TI?
e) Explain the second part of the TI-result using a self chosen example.
f) Can you explain where this strange second part is coming from?
                                   b−x    a+x    b   3x − a   b + 2a
                                      2
                                        +    2
                                               = 2 −     2
                                                            +
                                    a      b     a     b       a 2b 2




Example 3 (form 11)
Five spheres (R = 3.17 inch) made of brass are melted down and a cylinder with r = 6.34 inch is cast. What is
the length of the cylinder?

I had the students encouraged to compile their individual electronic collection of formulae stored on their
handheld CAS-devices. It was very interesting to observe that among the 18 students more than 10 various
ways appeared to solve this easy problem. The range reached from calculating like in secondary 1 to the
following elegant “One-liner”. Working with CAS encourages working with variables and with functions.

                          solve(5*sh_v(3.17)=cyl_v(6.34,l),l)

Example 4
Given is a set of solutions L = {3, -1, ½}
Find two equations of degree 5 with L = set of solutions.
I had not done this before, but the students knew about multiple solutions appearing in equations of higher
degrees.


Example 5
Find the solutions of the following equation without applying the solve()-command:
                         36x6 + 132 x5 - 263x4 - 638x3 + 926x2 - 770x + 1225 = 0



There was one partial project group with teachers testing alternative assessment forms (problem solving
tests, group tests, group presentations, “Facharbeiten”, …The very interesting report can be found in [11].




Josef Böhm: CAS in Austrian Classrooms                                                                        3
4         CAS and Standards
The question of establishing education standards in general and of standards for math education in particular
is in the focus of discussion in many countries. So it is in Austria, too. As a consequence of TIMSS and
PISA and of opening the borders for students in Europe we need comparability of the education systems.

Problems and questions for testing standards have been developed for Secondary 1 and a group of experts
with Helmut Heugl in a leading position are working on the standards for Secondary level 2. They are
considering including CAS into the standards according to the demands of our curricula.

The work of the team of experts is based on Heugl´s Competence Model. He says:

      CAS-supported mathematical education supports and encourages the 4 key qualifications
      •    Subject competence
      •    Methodological competence
      •    Social competence
      •    Personal competence
      much better than traditional mathematics education.

and then he continues about “the necessary fundamental algebraic competence in the age of CAS” addressing
competences for
      •    Recognizing structures
      •    Finding terms and formulae
      •    Testing
      •    Calculating
      •    Visualizing
      •    Modular working
      •    Finding the appropriate tool

Here is one test item for secondary 2:

1.   A test consists of 12 multiple choice questions with 4 answers for each of them. One answer is correct.
     The answers are marked randomly. X is the number of correct marked answers. The following diagrams
     show the distribution of X.
     a)    What is represented by the shaded area? Give your answer in words and in an appropriate
           mathematical form.




            umgangssprachlich (in words):                   umgangssprachlich (in words):



            mathematisch:                                   mathematisch:



Josef Böhm: CAS in Austrian Classrooms                                                                     4
The first test of the standard test items brought an unexpected side result. 21 classes of grammar schools
were tested (randomly chosen). It turned out that 10 of them were CAS classes. The evaluation showed that
the CAS-classes performed much better than the traditional educated classes. (All problems were solved
without any tool and they had not been designed considering CAS).

                                             gewichteter Gesamtindex

                     5
                                                                           4,33
                                      3,81                                 4,10
                     4
                                          3,33
                     3

                     2

                     1
                                                                                   7. Klasse
                                                                                   8. Klasse
                     0
                                    kein CAS                            CAS

This is encouraging and leads immediately to the next crucial point of discussion:
(It must be said, that 21 classes is not so much, to deduce a significant statement. Another result was that the
standard deviation of correct answered questions was greater in the CAS –Group.)


5       CAS and Basic Skills
One of the main obstacles which must be overcome by propagating CAS in math education is the fear of
teachers (and parents, too), that the so called “basic skills” might get loss. Question: “Which are Basic
Skills?” Read more about this in the “provocative” paper ([2])

                     Indispensable Manual Calculation Skills in a CAS Environment
    Wilfried Herget (Halle, Germany), Helmut Heugl (Wien, Austria). Bernhard Kutzler (Leonding, Austria) and
                                     Eberhard Lehmann (Berlin, Germany)


Many of us use CAS to train these „Basic (Manipulating) Skills“ up to a certain amount, because it is our
opinion that even in times of a CAS some manipulating skills should remain, because otherwise the pupils
cannot develop a certain “mathematic feeling” for solving problems supported by a powerful machine and
they will too much work by “try and error”.
We can provide the students with programs to train manipulation skills and we made the experience that
these training programs were highly appreciated by the students and by the colleagues.

This is a screenshot of a DERIVE program to train factoring expressions and polynomials:




The students are given randomly created expressions. They have to factor them using paper and pencil.
ch = presents the correct answers. (Here for three problems, but the number of problems can be varied).


Josef Böhm: CAS in Austrian Classrooms                                                                         5
Factoring random polynomials offers three options: According to the math level you can ask for factoring for
rational, irrational or complex roots:




The next screen shows how to train expanding expressions. Squaring and cubing binomials and calculating
other products can be done – and immediately by checked, accompanied by a kind of error analysis.




Similar program packages exist for the CAS-TI-devices
With "Catch the Points!" I am offering a „shooting game“. Randomly generated grid points are presented.
The task is to hit all points with as less straight lines as possible. Later we can repeat the game using
parabolas or other function types.




I´d like to underline that this is not the main purpose to use CAS, but sometimes a useful side effect.


Josef Böhm: CAS in Austrian Classrooms                                                                      6
Training of skills in differentiating and integrating (randomly generated problems of different levels of complexity).
The first row shows an example for applying the chain rule (level I) which was answered correctly, the second row
shows a problem for applying integration by parts (level I). After two false answers the system gives the correct
solution.
We know from our experience that students really appreciate tools like these and they practise much more
compared doing exercises from textbooks. They feel responsible for their success.

6       CAS and Dynamic Geometry
Fortunately geometry is becoming more and more important in our schools again. I am sure that the
availability of excellent dynamic geometry programs is responsible for this revival. (Cabri, Euklid, Geolog
and especially GeoGebra in recent times). What we are still missing is a dynamic geometry program which is
in a tight connection with CAS. Working with slider bars in DERIVE helps in a surprising way to narrow
this gap between algebraic and geometric representation form and can easily be done in classroom. Here is
one example:

It is well known that the parabola is defined as the locus of all points which have the same distances from a
fixed point and a fixed straight line. We generalize and ask for the locus of all points with a constant ratio of
these distances.


The figure shows the usual definition of a
parabola with the ratio mentioned above
e = 1. p is the variable distance between
the fixed point M and the fixed line l and
d is the variable radius of the circle with
its center in the fixed point M. Which is
the locus of points X with
e = MX : lX = const (= 1 in the figure).


We work in the Algebra Window with variables and each of the variables is represented by a slider bar in the
2D-Plot Window. Each step in the “Calculation Area” is followed by the parallel step in the “Plot Area”. So
we have a form of “Integrated Circuit” between abstract algebraic calculation and realistic geometric
visualisation.




Josef Böhm: CAS in Austrian Classrooms                                                                                   7
This is a part of the calculation performed in the Algebra Window:




And the result is surprising for all students (and for many teachers, too). We obtain one single formula for all
conics. (the Apollonian property of the conics). Many other examples (easier and much more complicated)
can be given. Changing the value for e we can obtain all forms of conics.


7       CAS and In-/Pre-Service Trainings
As I mentioned earlier the ACDCA was and still is one of the driving forces propagating CAS in classroom.
ACDCA is represented by members in all federal states. They have good connections to the Pedagogical
Institutes in their countries. Because of financial restrictions things have changed a bit. In some countries the
colleagues organize individual courses in the frame of their Working Groups. Things will change much more


Josef Böhm: CAS in Austrian Classrooms                                                                         8
in the future because of establishing the Pedagogical Universities and Mathematical Competence Centres.
We don´t know at the moment where the way will lead. It is an exciting time for all of us.

Same happened with the Pre-Service Courses. We had “Technology Courses” for our young teachers in all
federal states (Half day introduction for each: didactical use of Spreadsheet, Dynamic Geometry, Graphing
Calculator, CAS) supported by T3. At the moment I do know about one course in Lower Austria (organized
by the Pedagogical Institute) and one course in Tyrol (organized by Heiner Juen, P.I. Tyrol).



8       CAS and (Teacher-) Students
There are special lectures for teacher students on some universities (University of Vienna, Technical
University of Vienna, University Graz, University Klagenfurt as far as I do know) which shall introduce
working with technology in mathematics education – with a special impact on the change in the way of
teaching. The students are presented meaningful use of Internet, spreadsheet programs, graphing calculators
and CAS. In our lecture (TU Vienna) the students have to deliver a final paper in order to obtain a certificate.
(Lecture is held together with Gaby Bleier.)

Just recently I received a final paper on Sequences and Series including investigating monotony and
convergence. See what the student presented on the work sheet for pupils (among other examples):
                   1
    Let bn = 3 +        .
                   2n
    Show that for each arbitrary small number ε there exists an index n0 that bn − 3 < ε for all n ≥ n0.

This was the expected solution given by the student: (first for ε = 0.001, then for ε = 0.0001 and finally for a
generalized ε):




The student used CAS only as a calculation tool – but she added a verbal answer. This might be appropriate
in many cases. But CAS offers much more possibilities to do mathematics. Without a CAS the pupils had to
solve the inequality and I know from my experience that this is hard enough for many of them (considering
the absolute value, considering equivalence transformations in inequalities, …)

DERIVE offers the unique and unfortunately widely unknown and underestimated “Stepwise Simplification
Tool”. Why not ask the students (the classroom students) to simplify the solution process stepwise and to
explain each performed step. (Which rule applies? What is its purpose? What is the effect? …). Colleagues
using this tool in the right situation are very excited about the enrichment of teaching. (We include it in our
technology-oriented textbook, too.)

Using the “Steps” we can combine “Basic Skills”, Verbalising, Interpretation and other objectives of math
education in a wonderful way. Let´s follow the steps together with our students in the classroom and let´s
give comments on the rules which are lying behind the simplification process:




Josef Böhm: CAS in Austrian Classrooms                                                                        9
                                         If n>0,




Josef Böhm: CAS in Austrian Classrooms             10
Then we can generalize:



I am quite sure that even many (university) students and maybe also experienced teachers might face
problems in verbalizing the single steps.

Extending stepwise simplification was a main task for developing future DERIVE releases but unfortunately
by ending the support of DERIVE we cannot expect a wider range of stepwise simplifications. The (weak)
hope remains, that other CAS will follow this example!


9       CAS and Teachers
CAS in the classroom is impossible without teachers who like to work with CAS and who hopefully are
using CAS not only as calculation tool. Unfortunately there are also teachers – and as I heard just recently –
not only in Austria who use CAS to work through the traditional textbooks example for example supported
by powerful CAS-tools.

I admit that it is not easy to convince hesitating teachers to work with a CAS and I also admit that there are a
lot of good reasons to hesitate changing well established traditional teaching methods. I also admit that
teaching without CAS can be excellent math teaching and teaching with CAS can be very bad. Let me tell
one nice experience made in an in-service course. We built the model of a traffic density measurement for
one day. After some experiments we came up to superimpose two bell shaped functions and using the slider
bars we found a satisfying model function (for further investigations – differentiate, integrate, …)




Finally we wanted to extend this daily model for more days. What to do? After a while somebody brought
into discussion: ´”This is periodical! Trig functions?”. And after another while another colleague whispered
(very uncertain) “Fourier?”.

But nobody could remember how to perform a Fourier analysis because this is not in the curriculum of the
schools where the participants came from. So we opened the Online Help and searched for FOURIER.

Josef Böhm: CAS in Austrian Classrooms                                                                       11
This was the result after calling FOURIER(veh_dens(t),t,0,24,4), which was obtained in a part of
a second:




The next surprise was the bulky expression for the Fourier series (which I don´t want to copy!)

I want to tell the end of the story: One – very experienced and very traditional – teacher who had started the
course with many reservations became more and more enthusiastic when he saw how we used CAS. When
he saw the curve for the week he exclaimed: “That’s really great Josef, I am feeling like an apprentice of a
sorcerer (Zauberlehrling)”. The same happens sometimes in classroom – maybe not with FOURIER but with
other examples. Black Boxes can and should be used to make pupils (and teachers) curious about the
contents of the Black Boxes.


10      CAS and Textbooks
CAS in the classroom needs CAS in the textbooks. Many teachers complain that the textbooks don´t take the
use of CAS into account in the right way. If the textbooks make use of CAS then in many cases developing
the concepts and the problems don´t change, but there is an appendix – either at the end of the chapters or at
the end of the textbook – how to use CAS. (Mostly DERIVE and/or the TI). There is need for textbooks
which integrate CAS (and other technologies as well) to teach mathematics (the concepts), to encourage
experimenting, arguing, working on open ended questions in addition to a sound base of traditional
mathematics.
Three colleagues and I have been asked by a publisher to write a series of textbooks according to the
curriculum (which demands the use of technology). Three volumes are ready, volume 4 will appear in fall
2007. The textbooks are for Secondary Colleges for Business Administration.




The figure shows how students can make up their conjectures how to find the derivative of ef(x).

The next figure demonstrates the solution of a finance problem using the various tools. Working with CAS
we underline the use of self defined functions (here for present value, future value, etc.).

Josef Böhm: CAS in Austrian Classrooms                                                                     12
11      CAS and a new “Teaching and Problem Culture”




The above screenshot is a favourite procedure for many teachers using CAS
in a very early stage: compare useful and not useful equivalence
transformations of equations.

The right DERIVE-terms are one of (many) small examples to start
mathematics discussions. An absolute value being negative?? Impossible!!
Or not!! (Thanks to Walter Klinger for this nice problem.)

Many classroom examples from secondary I can be found in [1] where
special emphasis is given to experimenting, conjecturing and visualizing.

Josef Böhm: CAS in Austrian Classrooms                                      13
The following screens are very similar to the figures from paragraph 10 but not from the field of Calculus.
Using – and generating - random exponents the pupils should find out the rules for manipulating powers. We
find it important that in many cases the pupils are able to create the tools by themselves and they do not
depend on tools prepared by the teacher. Show them how to investigate multiplication and then they shall
find out how to investigate the remaining rules …




This is not the end of the story. The experimenting phase is followed by the exactifying phase which is the
proof for the conjecture(s).


Some examples for possible “new” questions are:

               a 2b
-   Rewrite         in at least 4 different ways.
              15c3
-   Fill in the form:
                         x 4 − 16 x 3 + 96 x 2 − ......

                         a 6b 3 + 6a 4b5 .......................................... = (...........................)3

                          For the next give more than one solution!!

                         .............. − 180 x 3 z ..................... = (..........................) 2

                         729 x12 − 5832 x10 y + 19440 x8 y 2 ........................................... =




Josef Böhm: CAS in Austrian Classrooms                                                                                 14
I am very often asked how to find problems for CAS-supported teaching. My answer is:




Compare the worked through traditional “edition” (next page) with the problems set in the new version. The
traditional version is a problem from a final exam from the 1980’s.


If you compare the two versions carefully you will find out that the new one does without asking for the
quartic. But there are some new questions and problems from a) to c) which emphasize visualization,
verbalisation, organisation of working.

Question f) addresses the Tool Competence.

Problem g) is not so easy because the 2nd derivative test gives a ”wrong” answer. One has to take the “sign”
of the area into account to give the right interpretation.




Josef Böhm: CAS in Austrian Classrooms                                                                     15
Josef Böhm: CAS in Austrian Classrooms   16
My personal recipe for designing “new” problems (which is often “reanimating” old problems):

-   Include the graphing capabilities for additional tasks - forcing visualization.
-   Encourage and ask for numerical and graphical solutions, ask for more than one way of solution, accept
    heuristic methods, encourage and ask for "Try and Error Methods".
-   Let the pupils produce self made tools and give opportunities to apply them.
-   Provoke functional thinking.
-   Find the problem to presented solutions.
-   Let the pupils set up conjectures - but also ask sometimes for the proofs.
-   Include the various ways of the CAS´ output into the task, interpret the output and compare with
    calculation by hands, make the CAS built-in simplification rules to a mathematical subject.
-  Use the CAS-capabilities or deficiencies for additional questions (and keep staying within the
   curriculum!!)
Another example:
Working with Linear Regression is standard – but is this question also standard??




We can find the regression line for the data points (using CAS).
We pick out one point – the red one (≈ (85,15). Moving this point in the plane results in different regression
lines. (This can be demonstrated by the use of slider bars). Some of the lines are parallel to the first one.
What is the locus of all (red) points which lead to regression lines with the same regression coefficient? Do
you know?

Or take this: Bring your environment into the math classroom.

                                                      This is the MAZDA-Logo. Can you describe the
                                                      form of this logo.

                                                      We use the slider bar for the exponent n:




                                                      n = 2.35. (It is a Lamé-Curve, Hyper-ellipse).

Following Group Work: Find and model other logos.



Josef Böhm: CAS in Austrian Classrooms                                                                      17
As I should focus on the use of CAS in our classrooms all other technologies have been pushed in the
background. We know that especially modelling can be done with numerical tools (spreadsheet) in an
excellent way, but it seems to be useful to do this also with CAS and to compare the modelling process.

See one example from Business Mathematics:

Supply- and Demand functions (a(p) and n(p)) are given (both dependent on price p(t)):

               a (p ) = 0 ,7 p 2 + p + 0 ,3 and n(p ) = 2 ,5 − 0 , 2 p − 0 ,15 p 2 ; p = p (t ); p (t = 0) = 1,00

The rate of change for price p is given by the following differential equation (dependent on time t):
                                                   p′(t ) = k ⋅ (n(t ) − a(t ))

Factor of proportionality k = 1 and the time interval ∆t = 0.1.

This is the respective difference equation (which can be treated with Excel):
                                ∆p
                                     = k ⋅ (n(t ) − a(t )) → ∆p = ∆t ⋅ k ⋅ (n(t ) − a(t ))
                                 ∆t
                                 p(t + ∆t ) = p(t ) + ∆t ⋅ k ⋅ (n(t ) − a (t ))
                                    pnew = pold + ∆t ⋅ k ⋅ (n(pold ) − a(pold ))

The difference equation on the CAS-TI (Sequence model)




The same with DERIVE (Iterations)




•   Possible question: which effect has a change of k?


Josef Böhm: CAS in Austrian Classrooms                                                                              18
Solving the differential equation with a Black Box:


The discrete points from before together with the numerical solution of the DE:




The differential equation on the TI:




Calculation of the equilibrium point:




Josef Böhm: CAS in Austrian Classrooms                                            19
A final example from just recently (source: my colleague and CAS-fighter Tania Koller, HAK St. Pölten)

Tania and her colleague found a strange (very traditional) example in a textbook:

Find a cubic with a turning point in (-3, 2) and an inflection point for x = -1 which passes P(3,7).

The respective system of equations had – surprisingly enough- no solution:

SOLVE([f(-3)=2, f'(-3)=0, f(3)=7, f''(-1)=0], [a,b,c,d])=[]

What happened? It was a typing error. But fortunately enough, the teachers and the “CAS-infected” students
were not satisfied with this answer. They changed one or the other value and then the system had a solution
and there was a resulting cubic. Then they generalized the problem:




and they found out, that exactly t = –1 causes the problems. Working with the slider bar supported their
results visually.

Having a CAS at their disposal they could go further. Why is –1 the bad boy/girl? Go on with generalizing:




The same denominator appears!




Factoring the denominator and leaving the trivial case xe ≠ xp, we find that xp ≠ 3xw – 2xe.
So we can design many tasks of the same kind which “don´t work”. Try to do this without a CAS!!

What did I say, some pages earlier: “Make New from Old!”

Josef Böhm: CAS in Austrian Classrooms                                                                     20
12       CAS and Visions
We all have visions for a PeCAS (Pedagogical CAS) or EduCAS (Educational CAS). I´d like to finish this
paper offering some ideas which I have collected in several talks with Austrian colleagues:

Main Visions are:

-    to make a CAS very intuitive to use. Teachers and students should not waste time in getting lost by
     considering technical details.

-    to develop a CAS-standard language (syntax) for a base set of commands (solve, simplify, plot,
     differentiate, integrate, …) together with at least partial compatibility between PC and handheld
     versions.

-    to integrate the various technologies which are important for math education: CAS and Dynamic
     Geometry, Spreadsheet, Statistics Package, Probes, Text processing, Programming, CAD, Portability
     into web formats.

-    to develop and to find new possibilities for teaching and learning mathematics which are based on the
     power of a CAS. One excellent example for this is in our opinion the Stepwise Simplification tool of
     DERIVE.

-    to find some agreement about basic knowledge, basic skills, basic concepts. Which amount is
     indispensable without CAS? See more in [2].

-    CAS should be used in the sense of a tool. Tools can and shall support and increase mathematics
     activities and insights.




Josef Böhm: CAS in Austrian Classrooms                                                                       21
13       References
This is a selection of papers. You can find many additional materials on the ACDCA-website
(http://www.acdca.ac.at) and many books on the website of bk teachware. (http://shop.bk-teachware.com)


[1]    ACDCA (Austrian Center for Didactics of Computer Algebra):
       Berichte der Forschungsprojekte,CAS III and CAS IV and Medienvielfalt.
       Homepage ACDCA: www.acdca.ac.at, www.geogebra.at, www.mathe-online.at
[2]    Herget, W., Heugl, H., Kutzler, B., Lehmann., E. (2000): „Indispensable Manual Calculation Skills in a CAS
       Environment“ in Ohio Journal of School Mathematics Autumn 2000, Number 42, Page 13. www.acdca.ac.at
[3]    Heugl, H., Klinger, W., Lechner, J. (2001): Mathematikunterricht mit Computeralgebra-Systemen. Addison-
       Wesley Publishing Company, Bonn 1996. ISBN 3-8273-1082-2,
       download from http://www.acdca.ac.at/material/allgem/buch/buch96.htm
[4]    Heugl, Helmut., (1999): The necessary fundamental algebraic competence in the age of Computeralgebra
       Systems. Proceedings of the 5th ACDCA Summer Academy, 1999,
[5]    Liebscher, Marlies. (2004): Projektbericht: „Bildungsstandards aus Mathematik für die Sekundarstufe II“.
       Projekt des BMBWK, Leitung Marlies Liebscher. CD-Rom des Landesschulrates für Steiermark.
[6]    Svecnik, Erich (2005): Projekt Bildungsstandards Mathematik Sekundarstufe II. Erste Ergebnisse der Erprobung
       von Aufgaben an AHS-Schüler/innen der 11. und 12. Schulstufe im April 2005. ZSE – Zentrum für
       Schulentwicklung, Abteilung Evaluation und Schulforschung, Graz-Klagenfurt
[7]    S. Fürst, SchülerInnenarbeitsheft zum TI/92, V200 für die Sek 1, bk teachware
[8]    J. Böhm, Neue Aufgaben für das Unterrichten mit DERIVE und TI-CAS-Rechnern, bk teachware
[9]    B. Kutzler, Lineare Gleichungen und Gleichungssysteme lösen, bk teachware
[10]   How to Make Traditional Tasks Technology Compatible,
       http://www.acdca.ac.at/material/allgem/howtomake.htm
[11]   New forms of teaching provoke and require new forms of assessment
       http://www.acdca.ac.at/material/allgem/assessment.htm
[12]   http://www.acdca.ac.at/material/t3/t3spin.htm
[13]   Elementarization and Modularization - two didactical aims being realized by using computer algebra systems
       http://www.acdca.ac.at/material/t3/t3modul.htm
[14]   A summary about the experiences how to integrate personal computers and hand computers (TI-89/92) in
       Mathematical Education in Austria
       http://www.acdca.ac.at/material/vortrag/sci2001.htm
[15]   Beispielsammlung / Collection of problems
       http://www.acdca.ac.at/material/bsp/index.htm
[16]   H.D. Hinkelmann a.o., Mathe mit Gewinn 1-4, oebv & hpt
[17]   Prugger a.o., Differenzialrechnung mit dem TI-92, bk teachware
[18]   Prugger a.o., Integralrechnung mit dem TI-92, bk teachware


Private communication with Gaby Bleier, Peter Hofbauer, Heiner Juen, Walter Klinger, Tania Koller, Josef
Lechner, Walter Wegscheider, Otto Wurnig, and others.



Author: Josef Böhm, email: nojo.boehm@pgv.at




Josef Böhm: CAS in Austrian Classrooms                                                                              22

								
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